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Simple Quantum Protocols and Algorithms-1

Quantum Key Distribution with BB84

Dr. Shreya Banerjee

08.07.2024

Center for Quantum Science and Technology

Siksha ‘O’ Anusandhan University, Bhubaneswar

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Quantum Secure Communication Protocol

A classical channel could be a communication wire like a telephone line where electrical signals represent bits or encoded information we send.

A quantum communication channel can be a fiber optic cable( as quantum cryptography is implemented using polarization scrambling) through which we can send individual photons (particles of light).

We will see how Alice and Bob will device a common key without ever telling each other what the key is.

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Two Different Measurement Bases

M-Z

M-X

The basis vectors for M-Z is |0> and |1>.

The basis vectors for M-X is |+> and |->.

Outcome of a single measurement: 0

Outcome of a single measurement: +/-

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Protocol

Alice generates a (classical) random string with 0 and 1s.

Say, ‘00110101’

From this string the key will be generated.

Alice and Bob both randomly selects two sequences of measurement bases, length of the sequence(s)= length of the string.
Say,

Alice’s sequence:: ZXXZXXZZ

Bob’s sequence :: XXZZXZZZ

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Alice’s encoding

String: 00110101

Sequence: ZXXZXXZZ

Alice takes 8 qubits.
Prepares them in computational basis (Z-basis):

Measures them according to her sequence: Applies

‘H’ gate when its X, does nothing when its ‘Z’.

Alice now sends her qubits to Bob.

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Bob’s Measurements

Bob’s sequence :: XXZZXZZZ

Bob measures the qubits of Alice’s state in X basis wherever there is an ‘1’ in his sequence, Rest he measures in Z basis.

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Key Generation

Alice and Bob publicly announce their bases.
They discard the positions where the bases do not match.

The positions that are not discarded, are kept and Bob uses his measurement outcome as key.
Let’s see how.

Position	1	2	3	4	5	6	7	8
Alice	Z	X	X	Z	X	X	Z	Z
Bob	X	X	Z	Z	X	Z	Z	Z
Outcome	NO	YES	NO	YES	YES	NO	YES	YES

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Measurement Outcome

Alice’s Bits	Encoding basis	Alice’s states	Bob’s Measurement basis	Bob’s outcome	Bob’s bit
0	Z	0	X	+ (probable)	discarded
0	X	+	X	+	0
1	X	-	Z	1(probable)	discarded
1	Z	1	Z	1	1
0	X	+	X	+	0
1	X	-	Z	0(probable)	discarded
0	Z	0	Z	0	0
1	Z	1	Z	1	1

KEY: ‘01001’

This is the famous BB84 Protocol for quantum key distribution.